Editing Hasse–Weil zeta function (section)

{{Short description|Mathematical function associated to algebraic varieties}}
In [[mathematics]], the '''Hasse–Weil zeta function''' attached to an [[algebraic variety]] ''V'' defined over an [[algebraic number field]] ''K'' is a [[meromorphic function]] on the [[complex plane]] defined in terms of the number of points on the variety after reducing modulo each prime number ''p''. It is a global [[L-function|''L''-function]] defined as an [[Euler product]] of [[local zeta function]]s.

Hasse–Weil ''L''-functions form one of the two major classes of global ''L''-functions, alongside the ''L''-functions associated to [[automorphic representations]]. Conjecturally, these two types of global ''L''-functions are actually two descriptions of the same type of global ''L''-function; this would be a vast generalisation of the [[Taniyama-Weil conjecture]], itself an important result in [[number theory]].

For an [[elliptic curve]] over a number field ''K'', the Hasse–Weil zeta function is conjecturally related to the [[Group (mathematics)|group]] of [[rational point]]s of the elliptic curve over ''K'' by the [[Birch and Swinnerton-Dyer conjecture]].